Multiple solution Optimization Strategy for Multi …epubs.surrey.ac.uk/848610/1/SMCA-18-04-0423-Clean.pdfMultiple-solution Optimization Strategy for Multi-robot Task Allocation Li

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Abstract—Multiple solutions are often needed because of

different kinds of uncertain failures in a plan execution process

and scenarios for which precise mathematical models and

constraints are difficult to obtain. This work proposes an

optimization strategy for multi-robot task allocation (MRTA)

problems and makes efforts on offering multiple solutions with

same or similar quality for switching and selection. Since the

mentioned problem can be regarded as a multimodal optimization

one, this work presents a niching immune-based optimization

algorithm based on Softmax regression (sNIOA) to handle it. A

pre-judgment of population is done before entering an evaluation

process to reduce the evaluation time and to avoid unnecessary

computation. Furthermore, a guiding mutation operator inspired

by the base pair in theory of gene mutation is introduced into

sNIOA to strengthen its search ability. When a certain gene

mutates, the others in the same gene group are more likely to

mutate with a higher probability. Experimental results show the

improvement of sNIOA on the aspect of accelerating computation

speed with comparison to other heuristic algorithms. They also

show the effectiveness of the proposed guiding mutation operator

by comparing sNIOA with and without it. Two MRTA application

cases are tested finally.

Index Terms—Multi-robot task allocation, multimodal

optimization, niching immune-based optimization algorithm,

Softmax regression, guiding mutation operator.

I. INTRODUCTION

ULTI-robot cooperative systems attract much attention

due to their distributed parallel processing abilities in

recent decades. They are considered to be promising and

of wide applications, such as path planning [1], exploration [2],

tracking [3], foraging [4], and transportation [5], and have been

This work was supported in part by the National Key Research and

Development Plan from Ministry of Science and Technology

(2016YFB0302700), National Natural Science Foundation of China (nos.

61473077, 61473078, 61503075, 61603090), International Collaborative

Project of the Shanghai Committee of Science and Technology (no.

16510711100), Shanghai Science and Technology Promotion Project form

Shanghai Municipal Agriculture Commission (no. 2016-1-5-12), the Fundamental Research Funds for the Central Universities (no. 2232015D3-32),

Cooperative research funds of the National Natural Science Funds Overseas

and Hong Kong and Macao scholars (no. 61428302), and Program for Changjiang Scholars from the Ministry of Education (2015-2019).

Li Huang, Yongsheng Ding, Yaochu Jin and Kuangrong Hao are with the

Engineering Research Center of Digitized Textile and Apparel Technology, Ministry of Education, together with the College of Information Science and

Technology, Donghua University, Shanghai 201620, China. Yaochu Jin is also

with the Department of Computer Science, University of Surrey, Guildford, Surrey GU27XH, United Kingdom. (e-mail: [email protected],

[email protected], [email protected], [email protected]). Mengchu

Zhou is with the Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07032, USA (e-mail:

[email protected]).

applied to many domains like social science [6] and industrial

engineering [7]. With the increasing demand for multi-robot

cooperation in complex applications, the significance of

multi-robot task allocation (MRTA), which is to determine an

efficient and intelligent task assignment to improve the system

performance [8], is recognized by more and more researchers

[9-13].

A single optimal allocation plan may be insufficient. First,

uncertain emergencies occur occasionally during actual

execution processes of multi-robot cooperative systems, such

that the predefined optimal task allocation schemes cannot be

realized as expected. In the light of tackling different types of

failures, outstanding work has been done to handle them. For

the contention of resources, the interference is modeled when

multiple robots use the same resource [14]. For dynamic

settings of uncertain costs, a probabilistic cost representation is

introduced to incorporate uncertainty and interdependency via

distributional models [15]. For the varying coupling

relationships, an ontology-based behavior modeling and

checking system is proposed [16]. However, these methods can

only cope with specific types of events. There is a lack of

methods able to handle all kinds of unexpected failures. Hence

a group of optimal assignments are highly desired, such that the

predetermined scheme can be switched to another one when it

cannot continue anymore. Second, precise mathematical

models and constraints with comprehensive considerations are

difficult to be summarized in some realistic application

environments. Nevertheless, decision-makers are surely

conscious of the feasibility of given solutions. Multiple

solutions are needed as candidates. Finally, for a redundant

circumstance, more robots are available for limited tasks. A

group of assignment schemes can all satisfy the task

requirement. Based on the above analysis, we conclude the

necessity to provide multiple solutions for an MRTA problem.

Multiple optimal solutions are common in industrial

manufacturing processes as well and have highly valued

practical applications. First, apart from uncertain failures of a

system, physical or energy constraints have impact on the

long-term execution of a unique optimum as well. Next,

multiple solutions with same or similar quality are important to

foster the robustness and flexibility of a system and are useful

for the sensitivity analysis of a problem. Lastly, multiple

optimal solutions, as calculated by a centralized controller, can

act as the prior information of a distributed multi-robot system

to guide cooperative work and decrease the chances of resource

conflict and system deadlock.

In general, this kind of problems belongs to the field of

multimodal optimization. Classical heuristic algorithms,

because of their rapid convergence to an optimum, likely a local

one, cannot be applied to multimodal optimization problems

Multiple-solution Optimization Strategy for

Multi-robot Task Allocation

Li Huang, Yongsheng Ding, Senior Member, IEEE, MengChu Zhou, Fellow, IEEE,

Yaochu Jin, Fellow, IEEE, and Kuangrong Hao

M


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directly. Yet as a base, they may properly be combined with

other techniques to deal with such problems well.

Commonly used algorithms include Genetic Algorithm (GA)

[17], Particle Swarm Optimization (PSO) [18], Ant Colony

Optimization (ACO) [19] and Differential Evolution (DE) [20].

The Big Bang-Big Crunch (BB-BC) algorithm is improved for

multimodal optimization because of its low computational cost

[21]. The Estimation of Distribution Algorithms (EDAs) are

improved to deal with multimodal one by considering their

advantages in preserving high diversity [22].

Niching methods are widely taken as ideal techniques to

maintain a balance between the exploration and exploitation of

multiple solutions [23]. Related researches are done from the

perspective of overcoming shortcomings and enhancing

performances of basic methods. A classical niching method

requires setting a niche distance threshold [24]. For this

problem, a Twin-space Crowding (TC) is introduced to build a

parameter-free paradigm to eliminate the effect caused by a

parameter value [25]. A distance-based Locally Informed

Particle Swarm (LIPS) optimizer is proposed in [18] to

eliminate the need for specifying any niching parameter and

enhance the fine search ability of PSO. A PSO algorithm using

a ring neighborhood topology, which does not require any

niching parameters, is described in [26]. In order to avoid the

time complexity caused by pairwise distance calculations, the

work [27] proposes a fast niching technique by introducing the

locality sensitive hashing, which is an efficient algorithm for

approximately retrieving nearest neighbors. An improved

information-sharing mechanism among individuals is

introduced in [28] to induce more stable and efficient niching

behavior, and a newly proposed parent-centric mutation

operator is combined with a synchronous crowding

replacement rule in [29]. Besides niching methods, new paths

to gain the multimodal optimization ability are established. For

example, in Gaussian Classifier-based Evolutionary Strategy

(GCES) [30], multimodal optimization problems are regarded

as classification ones, and the locations and basins of optima

are saved by using Gaussian mixture models. In the

cluster-based differential evolution [31], the clustering partition

is used to divide the whole population into subpopulations for

locating different optima.

An Artificial Immune System (AIS) is a parallel distributed

self-adaptive system including features of evolutionary

learning, pattern recognition and associative memory. Clonal

Selection Algorithm (CSA), which evolves multiple

populations simultaneously, is suitable for solving multimodal

optimization problems. A hyper-mutation operator appears in

CSA often. However, it is semi-blind and inefficient to handle

complex circumstances. To overcome such drawback, the work

[32] embeds Baldwinian learning and orthogonal learning in

CSA. Thus, the hybrid learning CSA can perform effectively

and robustly as expected. An immune-inspired affinity model is

studied for the container multimodal transport emergency relief

to schedule a multimodal transportation flow of the chain

efficiently and reliably [33]. A biological notion in vaccines is

introduced into AIS to lead antibodies to unexplored areas.

Consequently, the exploration can be promoted in a search

space for solving multimodal function optimization problems

[34]. A fusion of the immune network and predication performs

well in regulating local and global search and guiding the

determinate direction of local search, thus improving search

ability for dealing with multimodal problems [35].

Most of the research achievements inspired by AIS for

multimodal optimization are able to improve search ability,

convergence speed and solution quality. In spite of some work

like [33] caring about time efficiency as well, the existing

methods are based on the idea of accelerating search processes.

Considering characteristics of AIS and niching techniques, the

computational complexity remains at a much-the-same level. In

regard to the causes of the computational complexity, one of

them is an expensive fitness evaluation process in complex

environments and realistic applications, for example, our

concerned MRTA problem.

In this work, considering that the fitness evaluation process

is one of reasons causing much time consumption, Softmax

regression [36] embedded in a Niching Immune Optimization

Algorithm (sNIOA) is proposed to filter newly produced

antibodies via pre-judgment based on historical information

and reduce the number of individuals who enter an evaluation

process. The computational procedure can be accelerated

through the way of decreasing the time otherwise required for

fitness evaluation. Furthermore, a Guiding Mutation (GM)

operator is proposed to enhance the effective generation of new

antibodies in order to improve the search ability of sNIOA,

thereby resulting in its extension called sNIOA-GM. In

comparison with sNIOA, the proposed sNIOA-GM is able to

search in a larger space while requiring less computational

resource. In simulation experiments, values of parameters are

analyzed for better performance; comparisons with other

multimodal optimization algorithms on benchmarks are

conducted; and two scenarios of MRTA problems containing

multiple solutions are tested.

The rest of this paper is constructed as follows: Section II

presents the proposed sNIOA. Section III introduces the GM

operator and sNIOA-GM. Section IV analyzes the multimodal

optimization ability of sNIOA and influences of its parameters

on its performance and compares it with other heuristic

algorithms. It also presents the performance of the GM operator

and shows task assignment results for MRTA problems. The

conclusion and future work are given in Section V.

II. NICHING IMMUNE-BASED OPTIMIZATION ALGORITHM

BASED ON SOFTMAX REGRESSION

The flowchart of the proposed sNIOA is shown in Fig. 1.

First, initial antibodies are generated according to an antigen

recognized, which represent initial solutions for a problem.

Their fitness values are calculated through an evaluation

process to determine affinity to the antigen. Then, these

antibodies and their fitness compose the training set of Softmax

regression. A pre-judgment model is built by it. New

population, produced by operators of selection, mutation and

crossing, is prejudged by this regression model such that

non-ideal (low-quality) antibodies can be eliminated before

entering an evaluation process. In this way, the time of fitness

evaluation can be reduced. The niche technology based on a

niching distance is used to maintain distribution and

multimodal performance of solutions.


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Initialize antibody

and training set

Whether meeting the

terminal conditions?

Produce

memory cell

Build pre-

judgment model

Y

N

Recognize

antigen

Output

Evaluate

antibody

Niche

techonology

Update

sample library

Generate parent

population

Produce new antibodies

（Selection, mutation

and cross)

Antibody pre-

judgment

Generate new

population

Fig. 1. The flowchart of sNIOA.

A. Immune-based Optimization Algorithm

AIS is described as the one that incorporates many properties

of natural immune systems, including diversity, distributed

computation, error tolerance, dynamic learning and adaptation,

and self-monitoring [37]. Therefore, it offers effective and

efficient inspiration for complex optimization problems. Based

on the mentioned properties, a series of intelligent algorithms

have been proposed. Their typical representative is an

immune-based optimization algorithm (IOA).

IOA is a combination of algorithms based on an immune

self-regulation mechanism and immune response [38]. It shares

similarities with other immune algorithms and possesses

unique and distinctive characteristics of keeping population

diversity and converging to the optimum rapidly. Attributing to

them, IOA is a preferable approach for multimodal

optimization. Its implementation procedure is shown in Section

A of the Supplementary File.

First, aiming at a recognized antigen, which corresponds to

an objective function in an optimization problem, initial

antibodies are generated randomly.

Then, individuals enter an evolution procedure based on

affinity values with the antigen and keep diversity by

considering concentration values among antibodies. This is an

application of the immune system antibody concentration

regulation principle, i.e., the antibody with worse values of

fitness as well as concentration is suppressed during the process.

Thus, the population diversity can be preserved along with the

reservation of matched antibodies. More specifically, the

evaluation phase is based on three indexes of individuals. The

first one is fitness, which corresponds to the value of an

objective function in optimization. The second one is

concentration, which indicates population diversity. The third

one is excellence, which is calculated based on their fitness and

concentration:

1 1

1i i

i e e

i i

f c

e p pn n

f c

i i

I II I I

I I

, (1)

where if

I , icI and

ieI denote values of fitness, concentration

and excellence of individual i , respectively; and epI is the

multiplicity evaluation parameter to balance fitness and

concentration as a weight.

Next, excellent antibodies constitute the parent population

and the best part of these is kept in the memory cell. The

memory unit helps the corresponding antibodies for an antigen,

which has come up, to be generated faster than before.

After that, new population is generated through three phases.

The first one is selection that is based on the roulette selection.

The individual with a greater value of excellence is chosen with

a higher probability. The second one is mutation used to

produce new individuals. It can search more space. The third

one is crossing that contributes to reserving gene segments of

good individuals and passing them on to future generations.

Two terminal conditions are used in this work. One is the

limited number of iterations. The other is that the average

fitness value of individuals reaches a predetermined threshold.

B. Niche Technology

The niche technology is an abstraction from a biological

phenomenon. In an ecological environment, fierce

competitions among intraspecific creatures may occur

sometimes for limited resources. However, such a relationship

does not appear among different species. In this way, in the

process of population evolution, the fittest one can be preserved

in a certain kind of creatures. At the same time, the species

diversity can be maintained. The introduction of a niche

technology to IOA is able to equip IOA with multimodal

optimization ability while maintaining diversity of solutions.

There are a variety of ways to implement niche techniques.

This work uses the one based on niche distance. First, a niche

distance threshold is defined. Then, the distance between each

pair of individuals is calculated. If the calculated distance is less

than the threshold, the individual that has worse fitness is

inflicted with a penalty. Thus it may be eliminated with a larger

probability later on. In other words, there is only one

satisfactory solution within the range of some niche distance.

The pseudo-code of this niche technology based on niche

distance for a maximization problem is shown in Section B of

the Supplementary File.

The niche distance is an important parameter of the niche

technology and has great impact on multimodal optimization

performance. A large niche distance may lead to

non-convergence of an algorithm. On the contrary, a tiny one

can result in non-uniformity of solutions, especially for the

function with continuous solution intervals, even missing the

optima.


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C. Softmax Regression

At the beginning of sNIOA, antibodies are initialized and

their fitness values are computed. These values are contiguous.

Hence, a fitness threshold should be given in order to replace

fitness values by class labels. The fitness threshold t is

called classification threshold and defined as:

ˆ ˆˆ ˆ , maximization

, minimization

f f f for

f f f for

, (2)

where f is the fitness value of an antibody and is the

parameter of a classification threshold. For maximization

problems, an antibody whose fitness is more than , is

expected to be preserved. For minimization problems, an

individual whose fitness is less than , is a better choice. In

our concerned MRTA problems, the individuals with better

fitness values are regarded as ideal ones. Others are called

non-ideal antibodies. After being labeled, the initialized

individuals and corresponding labels are input to a Softmax

training process as a sample database and then a prediction

model is obtained.

During iterations, when new antibodies are produced after

selection, mutation and crossing operations, their qualities are

determined via the mentioned prediction model. The non-ideal

(low-quality) individuals are eliminated directly. The ones with

predicted high qualities enter an evaluation process to obtain

their precise fitness values.

The newly computed antibodies and their labels are

embedded into the sample library of Softmax regression. The

classification threshold is recomputed, and the individuals are

relabeled with the new threshold to raise the standard of better

individuals gradually. The description of Softmax regression is

given in Section C in the Supplementary File.

III. GUIDING MUTATION OPERATOR FOR SNIOA

In this section, the GM operator is proposed to strengthen the

optimization ability of sNIOA, which is inspired by the theory

of gene mutation [39], thus yielding sNIOA-GM. In accordance

with the mutational pattern of a base, gene mutations can be

divided into frameshift mutations and base substitutions. The

former is caused by the insertion or deletion of a number of

nucleotides. For example, a segment of genes “A T C T”

changes to “A T A C T” by the insertion of “A” or “A T T” by

the deletion of “C”. The latter therein refers to a kind of

mutations that one base pair is replaced by another different

base pair. The GM operator is inspired by base substitutions.

Many of base substitutions originally derive from the

mutation of a single base. When a single base is changed,

another base in this base pair is also replaced by others in the

next gene replication. For example under the influence of

nitrites, a cytosine (C) mutates into a uracil (U) stimulated by

the oxidative deamination. During the gene replication, the

uracil (U) does not pair up with a guanine (G) but with an

adenine (A). As a result, the C-G base pair mutates into a T-A

base pair as shown in Fig. 2.

C

G

U

G

U

A

C

G

U

A

T

A

Fig. 2. Example of base substitution under the influence of nitrites.

With the inspiration mentioned above, dividing one antibody

into several “gene pairs” is taken into consideration. A

traditional gene pair consists of two bases. Herein, the number

of genes in a “gene pair” may be more than two. In other words,

the wording of “gene group” should be more exact. When a

certain gene mutates, the other genes in a same group have a

higher probability of mutation.

The definition of a gene group also makes sense to our

MRTA problem. For example, a proper group among many

available robots has been chosen but with the inferior

connections with tasks as shown in Fig. 3(a), where triangles

stand for tasks and circles represent robots. In this case, it is less

helpful if there is only one assignment line being changed as

shown in Fig. 3(b). When these tasks are regarded as a gene

group, others have a higher probability of mutation when one of

them is altered. Thus the combination has more chances to be

modified comprehensively as shown in Fig. 3(c).

(a) (b) (c)

Fig. 3. Illustration of the gene group in a MRTA problem.

In the above-mentioned scenario, the formation of a gene

group mainly depends on the locations of tasks. In other words,

tasks which are relatively close to each other tend to be in a

same gene group. So, the Mean-shift algorithm is used to do

this herein. Some examples are shown in Section D of the

Supplementary File. Different kinds of symbol shapes

represent different gene groups. The results of clusters can

adapt to different application demands by tuning the threshold

in the Mean-shift algorithm.

After the creation of gene groups, individuals are determined

to be whether mutated or not depending on mutation probability

mp . In the process of mutation, when one gene of an individual

is chosen to be changed, each of the rest of genes in the same

group continues to be decided with a higher mutation

probability hmp . The GM operator is summarized as shown in

Algorithm I.

Based on above discussions, the procedures of sNIOA-GM

for MRTA problems are proposed as shown in Algorithm II.

Given population size N , sample library size S and gene

group size G , the GM operator costs O N G O N ,

where G ≪ N . The niche technology takes 2O N . The

training and predicting processes of Softmax regression spend


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O S and O N , respectively. In summary, the proposed

sNIOA-GM’s complexity is 2O N O S .

Furthermore, the entire algorithm’s complexity depends on

the objective function being optimized as well. If it is a simple

function, it takes O N to evaluate the population. However,

when it comes to a complex context, especially appearing in

real applications, the evaluation process is complicated. When

its complexity exceeds O S , the proposed algorithm can

exhibit its great advantage than those without using Softmax

regression.

ALGORITHM I. GUIDING MUTATION OPERATOR

1: Base groups creation;

2: For each individual in population, generate a random number r ;

3: If mr p

4: Choose one base randomly and activate corresponding base group;

5: Base mutation;

6: For each of the rest of the base group, generate a random number r ;

7: If hmr p

8: Base mutation;

9: End if 10: End for

11: End if

12: End for

ALGORITHM II. SNIOA-GM

Step 1: Generate initial solutions according to the problem statement. Calculate

fitness values as the initial training set. Step 2: Use the niche technique to obtain multimodal optimization

performance.

Step 3: Evaluate the excellence ie of each ideal individual by the combination

of fitness if and concentration ic through a multiplicity evaluation

parameter ep . Stop if the termination criterions are met.

Step 4: Update the training set and create the prediction model. Step 5: Preserve good solutions in the memory cell.

Step 6: Generate new population through the way of selection, GM operator

and crossing. Step 7: Use Softmax regression to eliminate non-ideal solutions. Others are

preserved and combined with the memory cell as the parent

generation. Step 8: Increase the iteration count by one and go to Step 2.

IV. EXPERIMENTAL RESULTS AND ANALYSIS

In this section, the multimodal optimization ability of the

proposed sNIOA is shown by comparing with other popular

heuristic algorithms. This work provides the simulation results

of sNIOA-GM for two MRTA application cases.

A. Multimodal Optimization Ability of sNIOA

In this phase, the multimodal optimization ability of sNIOA

is tested to ensure the existence of alternative solutions.

Schaffer’s F6 function is a typical multimodal function

described by

2 2 2

22 2

sin 0.5, 0.5

1 0.001

x yf x y

x y

. (3)

In order to make the solution space including multiple

equivalent optimal areas, 2 20.001 x y is set to 0. That is to

say, the oscillations of function do not attenuate and each of the

peaks is a continuous interval. With the purpose of a clear

observation, this optimization problem is considered to search

for its maxima. The range of x and y is [-4, 4].

A group of random initial solutions are shown in Fig. 4(a).

The optimization results of IOA and sNIOA are shown in Fig.

4(b) and Fig. 4(c), respectively.

(a) Solution space of test function and initial solutions

(b) Optimal solutions of IOA

(c) Optimal solutions of sNIOA

Fig. 4. Illustration of multimodal optimization ability.

According to Fig. 4, both IOA and sNIOA can find the

optimization solutions of the revised Schaffer’s F6 function.

The result obtained by IOA, however, converges to one optimal

solution. To the contrast, obtained solutions of sNIOA can

distribute over the entire optimal solution intervals uniformly.

The theoretical reason is that, when an excellent antibody is

once generated, other antibodies are influenced by this

excellent one through the crossing operation during iterations.

Thus, they all converge to the excellent one in the end of IOA.

In sNIOA, a niche distance is defined. Because of this, there is a

fixed distance between each pair of candidate solutions, such

that solutions cannot gather to one point only. The training and

predicting processes, which are introduced to reduce the

evaluation time in sNIOA, do not have an effect on this

performance. To sum up, the uniform multimodal optimization

ability of sNIOA can be guaranteed.

B. Analysis of Parameters in sNIOA

The niche distance is a significant parameter of sNIOA. If it

is too small, it is difficult for sNIOA to find all optimal

solutions and leads to a premature convergence. On the other

hand, large niche distance is unfavorable for functions that have

steep peaks. In addition, the uniformity of optimal solutions in

the same continuous interval is also related to the distance

parameter.

The classification threshold as a parameter also impacts


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optimization results. A loose threshold lets more ordinary

antibodies enter the evaluation process such that the advantage

of a fast computational speed cannot be shown dramatically. In

contrast, a tight threshold may impose restrictions on

population diversity.

These two parameters mentioned above have great influence

on optimization performances of sNIOA. In this part, the

impacts of niche distance and classification threshold are

discussed through tests on the Branin function. Their values

and resulting experimental results are shown in TABLE 1.

TABLE 1 PARAMETERS ANALYSIS EXPERIMENTAL RESULTS

Parameters Experimental Results

n

Deviation

of

Optimal Value

Deviation of Mean

Value

Standard

Deviation

Evaluation

Time

Actual Peak

Number

0.02

1/12 1.64E-03 0.0541 0.0309 22739.6 3

1/3 7.38E-03 0.1884 0.1192 14616.2 3

2/3 4.96E-03 0.2953 0.1744 9018 3

11/12 5.80E-03 0.0696 0.0397 10635.4 3

0.05

1/12 1.78E-03 0.1337 0.0963 22432.4 3

1/3 5.30E-03 0.2493 0.1530 16547.2 3

2/3 7.20E-03 0.3719 0.2196 10296.2 3

11/12 6.08E-03 0.2028 0.1179 10308 3

0.1

1/12 1.98E-03 0.2493 0.1739 22287.4 3

1/3 5.20E-03 0.4675 0.4395 15172.2 3

2/3 1.48E-03 0.5104 0.3103 10473.2 3

11/12 1.25E-03 0.3817 0.2861 11317.4 3

0.2

1/12 3.78E-03 0.8018 0.5330 22890 3

1/3 4.50E-03 0.9707 0.6108 15034.6 3

2/3 5.10E-03 1.1528 0.6932 11694.4 3

11/12 7.48E-03 1.0844 0.7796 9778 3

According to TABLE 1, all the experimental combinations

can find all optimal peaks. The deviations of optimal value

approach to 0 almost, i.e. sNIOA can find all optimal solutions

of this test function and achieve the optimization goal.

Furthermore, we use the prior computation time of NIOA as a

reference, which is 24080 evaluation times. From TABLE 1,

the times of fitness evaluation of all experimental combinations

are less than 24080. It proves that sNIOA can reduce the

evaluation time and show a satisfied computational

performance.

Next, the effect of classification threshold is analyzed.

According to above discussions, this parameter mainly has

impact on the time of computation. This conclusion is verified

by Section E in the Supplementary File, that the evaluation time

decreases with the raise of classification threshold parameter.

The reason for this is that, in the process of Softmax training for

minimizing problems, an individual is regarded as a preferable

antibody when its fitness is less than . When increases in

the experiments, goes down, i.e. only antibodies whose

fitness is much smaller than others can be preserved at the end

of Softmax prediction. In consequence, fewer individuals go to

the evaluation process, such that the evaluation time can be cut

down drastically, e.g., 62.55%. The bigger value of this

parameter, however, may not be better. When this value

increases to more than 2/3, the evaluation time has a slight drop

or remains unchanged till the end and even increases in some

cases.

The niche distance mainly has effect on the deviation of

mean value and standard deviation of optimal solutions. Along

with the niche distance’s growth, both deviations are on the rise

as shown in Section F of the Supplementary File. The

explanation of this trend is that, when the niche distance

increases, optimal solutions are more scattered around the

peaks. Therefore, a bigger deviation is caused. Nevertheless, a

smaller niche distance may lead to non-uniformity of solutions,

especially for the problem which has continuous solution

intervals, even missing optima sometimes.

To sum up, for different problems, the specific combination

of parameters should be designed accordingly.

C. Comparison with other heuristic algorithms

In order to prove its efficiency, we compare sNIOA with

other popular heuristic algorithms on test functions next.

These algorithms are all proposed for solving multimodal

optimization problems including NIOA, Niching Genetic

Algorithm (NGA) and Niching Particle Swarm Optimization

(NPSO). They are selected because of their validated

performance [40-42] and use of the niching technique. The

parameters in each algorithm are shown in TABLE 2, where

N is population size; L is the length of binary coding; I is

the number of iterations; M is the capacity of memory; cp

and mp are probabilities of crossing and mutation, respectively;

s is the parameter that balances fitness and concentration of an

antibody; and are parameters of classification threshold

for maximization and minimization problems, respectively; n

is the niche distance; max and min are the ranges of the

inertia weight; and 1c is the learning factor.

TABLE 2

PARAMETERS IN COMPARISON EXPERIMENTS OF EACH ALGORITHM

sNIOA NIOA NGA NPSO

N 100 N 100 N 100 N 100

L 22 L 22 L 22 L 22

I 200 I 200 I 200 I 200

M 20 M 20 cp 0.9 max 1

cp 0.9 cp 0.9 mp 0.05 min 0

mp 0.3 mp 0.3 - - 1c 2

s 0.15-0.95 s 0.15-0.95 - - - -

2

3 n 0.01-0.1 - - - -

1

3 - - - - - -

n 0.01-0.1 - - - - - -

All parameters are selected carefully to ensure that each

algorithm performs well. For example, the mutation probability

mp appears both in sNIOA, NIOA and NGA. However, the

one used for NGA is 0.05 and is much less than 0.3 used in

sNIOA and NIOA. We try several values of the mutation

probability and choose the one that leads to the best

performance of each algorithm by testing them on the revised

Schaffer’s F6 function. The explanation can be offered from the

theoretical perspective. In NGA, since the final goal is to bring

the population to convergence, selection and crossing happen

very often. Mutation, attempting to occasionally break a

number of individuals out of local optima as a way to maintain


7

diversity, should occur less frequently. In sNIOA and NIOA,

there is a memory cell preserving well-behaved individuals in

each generation inspired by AIS. Thus the excellent solutions

are not eliminated in the following iterations when increasing

the probability of searching more areas. As a result, a large mp

is preferred.

Test functions are shown in TABLE 3. Within specified

limits, the revised Schaffer’s F6 function includes multiple

continuous global optimization intervals. Himmelblau function

and Branin function have multiple global optimal peaks

without local optima. Rosenbrock function is actually a

unimodal function with a continuous gradual interval beside the

peak. There are two optimal solutions and four local optima in

Six-Hump Camel Back function. Cross-in-Tray function and

Rastrigin function include four and one global optimal peak,

respectively, both with many local optima. Shubert function

includes two global optimal peaks, a small number of local

optima and many slight fluctuation peaks. Note that when the

minima of a function are negative, we move this function to

upward. Thus, the lower boundaries are 0 for all test functions.

These functions cover various kinds of characteristics of

multimodal functions, such as continuous intervals and steep

peaks, global optima with and without local optima and

maximization and minimization problems. They are also

chosen as commonly-used benchmarks by many researchers

[42-44]. Therefore, we use them as our benchmarks.

The experimental results keep a record of the deviation from

the optimal value, deviation from mean value, standard

deviation, evaluation time and number of peaks found as shown

in TABLE 4. Here the deviation from the optimal (mean) value

is the difference between the optimal (mean) values obtained

from optimization algorithms and theoretical optima. The

standard deviation is used to measure stability of solutions. The

evaluation time indicates the time performance. The number of

peaks found is applied to evaluate the multimodal optimization

ability. In addition to the number of iterations, the termination

condition of each algorithm also takes the deviation from the

mean value into consideration. In this way, meaningless

calculation does not proceed when the optimal value cannot be

improved any more during iterations. All of the data are mean

values of ten times repeated trials.

From this table, the excellent time performance of sNIOA

can be seen in comparison with other heuristic algorithms. The

proportion of evaluation time reduces on average by 41.01%,

38.38% and 39.34% compared with NIOA, NGA and NPSO,

respectively. Its maximum time reduction can reach 53.78%,

68.74% and 69.63% in comparison with its three peers. Its other

performance measures fall in normal ranges of optimization

results. Some of its performances are the best. It is easy to

conclude that sNIOA can speed up the computation

dramatically while showing comparable optimization

performance in comparison with its three peers.

From the point of view of multimodal optimization ability,

sNIOA is able to find all optimal peaks. Antibodies do not

converge to local optima in most of the functions due to the

repulsion of the niche distance, except Rastrgin function with a

large number of local extremums. However, the limitation of

sNIOA cannot be denied, i.e., the non-uniformity of individuals

is evident when peaks are steep. The experimental results show

that NGA has strong convergence ability. Because of this,

NGA’s ability to search unknown optimal peaks is reduced as

shown in Himmelblau and Branin functions in TABLE 4. The

search ability of NPSO is uncertain and unstable. It sometimes

results in convergence to local optima as shown in Branin,

Six-Hump Camel Back and Shubert functions.

From the perspective of three deviation values, sNIOA

performs noticeably well in the experiments of revised

Schaffer’s F6, Branin and Shubert functions. Other results are

of the same order of magnitude and approach to the optimal

values.

TABLE 3

TEST FUNCTIONS AND PARAMETERS

Test Function Equation Range Max Min

Revised Schaffer’s F6

2 2 2, sinf x y x y -4 , 4x y 1 0

Himmelblau 2 2

2 2, 11 7f x y x y x y -6 , 6x y 2200 0

Branin

22

2

5.1 5 5 5 1, 10 6 10 1 cos 5 10

4 8

x xf x y y x

-10 , 10x y 250 0

Rosenbrock 2 22, 100 1f x y y x x -2 , 3x y 2200 0

Six-Hump

Camel Back

42 2 2 2, 4 2.1 4 4

3

xf x y x x xy y y

-3 , 3x y 5 0

Cross-in-Tray

2 20.1

100

, 0.0001 sin sin 1

x y

f x y x ye

-4 , 4x y 1 0

Shubert

, 1 cos 2 1 2cos 3 2 3cos 4 3 4cos 5 4 5cos 6 5

1 cos 2 1 2cos 3 2 3cos 4 3 4cos 5 4 5cos 6 5

f x y x x x x x

y y y y y

-2 , 2x y 350 0

Rastrgin 2 2, 20 10cos 2 10cos 2f x y x x y y -5 , 5x y 80 0


8

TABLE 4

EXPERIMENTAL RESULTS ON TEST FUNCTIONS OF DIFFERENT HEURISTIC OPTIMIZATION ALGORITHMS

Test Function Optimal

Value

Optimization

Algorithm

Deviation of

Optimal Value

Deviation of

Mean Value

Standard

Deviation

Evaluation

Time

Actual (Ideal)

Peak Number

Schaffer’s F6 1

(max)

sNIOA 1.69E-08 9.59E-05 1.08E-04 6195 5(5)

NIOA 1.76E-08 9.51E-05 1.19E-04 7163 5(5)

NGA 1.90E-13 4.81E-04 1.76E-03 19820 5(5)

NPSO 0 1.27E-02 7.76E-02 20400 5(5)

Himmelblau 0

(min)

sNIOA 1.70E-03 1.3200 0.9378 11553.9 4(4)

NIOA 4.54E-04 0.9820 0.6493 24080 4(4)

NGA 3.20E-04 1.0500 0.7300 20100 3.5(4)

NPSO 3.07E-03 4.0534 3.9987 20400 4(4)

Branin 0.397887

(min)

sNIOA 3.27E-03 0.2525 0.1581 14346.1 3(3)

NIOA 1.77E-04 0.3044 0.2049 24080 3(3)

NGA 5.44E-05 0.2553 0.1686 20100 2.8(3)

NPSO 2.58E-03 1.3908 0.9989 20400 4(3)

Rosenbrock 0

(min)

sNIOA 7.72E-03 0.2574 0.1508 14036.2 1(1)

NIOA 6.14E-02 0.3093 0.0894 24080 1(1)

NGA 1.06E-02 0.2066 0.1431 20100 1(1)

NPSO 5.50E-04 0.4576 0.4708 20400 1(1)

Six-Hump

Camel Back

-1.0316

(min)

sNIOA 1.21E-03 0.0953 0.0655 12280.5 2(2)

NIOA 8.61E-05 0.0739 0.0509 24080 2(2)

NGA 1.97E-05 0.0275 0.0185 20100 2(2)

NPSO 5.69E-04 0.4158 0.3252 20400 3.8(2)

Cross-in-Tray -2.0626

(min)

sNIOA 4.29E-05 2.93E-03 2.03E-03 11130.1 4(4)

NIOA 1.19E-06 2.05E-03 1.33E-03 24080 4(4)

NGA 1.86E-06 1.08E-03 8.33E-04 20100 4(4)

NPSO 1.71E-05 1.23E-02 1.54E-02 20400 4(4)

Shubert -186.73

(min)

sNIOA 6.81E-02 7.8764 5.0709 14071.5 2(2)

NIOA 1.18E-03 8.4319 5.2265 24080 2(2)

NGA 3.32E-04 12.6556 8.5062 20100 2(2)

NPSO 1.65E-01 106.8010 52.1314 20400 15.5(2)

Rastrgin 0

(min)

sNIOA 7.53E-02 3.2800 1.2984 15389.1 14.6(1)

NIOA 1.94E-03 2.9933 1.2430 24080 13.5(1)

NGA 5.53E-09 2.2137 0.8807 20100 9(1)

NPSO 4.52E-01 6.8802 3.8715 20400 27.4(1)

Next, we illustrate if there are significant differences

between sNIOA and other heuristic algorithms. The significant

difference p value at a two-tail t-test is given with the

significance level of 5%. The null hypothesis is that there are no

significant differences between sNIOA and other algorithms.

As the results of four indexes shown in TABLE 5 for the

revised Schaffer’s F6 function as an illustrative example, if the

p value is smaller than 0.05, the null hypothesis is rejected, i.e.

there is a significant difference between two algorithms on the

corresponding index. The four indexes are divided into two

parts. The first part describes the optimization ability, which

consists of the deviation of optimal value, the deviation of mean

value and the standard deviation.

TABLE 5

T-TEST RESULTS ON SCHAFFER’S F6 FUNCTION

Compared

Optimization

Algorithms

p value

Deviation

of Optimal Value

Deviation of

Mean Value

Standard

Deviation

Evaluation

Time

sNIOA vs NIOA 0.95554 0.66327 0.10946 0.040021

sNIOA vs NGA 0.15213 0.0043328 0.037891 3.4412e-19

sNIOA vs NPSO 0.15214 5.1572e-05 1.3226e-07 6.4315e-13

As shown in TABLE 5, some results obtained from sNIOA,

for example the deviations from the mean value of sNIOA and

NGA, show significant differences. However, accompanying

with results shown in TABLE 4, the corresponding index of

sNIOA is better than NGA’s. Therefore, a conclusion can be

obtained that sNIOA has at least the same optimization ability

as other heuristic algorithms do. The second part describes the

time performance represented by the evaluation time.

Regarding this index, sNIOA performs the best on computation

speed as shown in TABLE 4. By means of the hypothesis

testing, significant differences are verified between sNIOA and

other algorithms as shown in TABLE 5. To sum up,

considering all experimental results synthetically, sNIOA

achieves the original intention of this work with competitive

multimodal optimization performances, and clear advantage on

computation speed over its three peers.

Next, two application cases are provided to illustrate the

merits of sNIOA-GM in solving MRTA problems.

D. Application Case 1

Assume that there are q robots 1 2, , , qR r r r and s tasks

1 2, , , sT t t t . At a certain moment each task needs to be

executed by one robot and each robot can only perform one task.

ij is a binary variable and presents the allocation relationship

between robot ir and task jt . ijc is the execution cost of task

jt by robot ir . Denote the priority level of task jt as jp . A

smaller jp implies a higher priority of task jt . The MRTA

problem can be stated as follows.

1

mins

ij ij j

j

f c p

(4)

s.t. 1

1, 1,2, ,s

ij

j

i q

(5)


9

1 1

, 1,2, , ; 1,2, ,q s

ij

i j

s i q j s

(6)

s q (7)

0,1 ,ij i j (8)

In this scenario, ten groups of simulations are done with the

number of tasks increasing from 10 to 100 and the number of

robots increasing from 20 to 200. The tasks have different

priority levels from 1 to 5. Tasks and robots are dispersed over

a 100×100 square area randomly. All the experimental

combinations are repeated for 10 times. Then average values

are taken as the final results.

The performances of sNIOA-GM are discussed from two

perspectives: run time and obtained cost, respectively. For the

former, due to the pre-judgment of Softmax regression, sNIOA

and sNIOA-GM both show better time performance than the

original IOA as shown in Fig. 5(a). Regarding the cost, the GM

operator can account for the elimination of second half

iterations of sNIOA-GM as shown in Fig. 5(b). When the

number of tasks is small, optimal solutions can be found under

the same conditions by all of these three methods. However

when it comes to large-scale task allocation problems, in the

case of same limited number of iterations which is 200, the GM

operator can make antibodies mutate toward a better direction

and help find the preferable solutions, while solutions given by

IOA and sNIOA cannot converge to optimal values.

In closing, sNIOA-GM can realize effective optimization in

the concerned MRTA scenarios by achieving rapid

computation speed and better solutions, especially for

large-scale problems.

(a) Run time

(b) Value of cost function

Fig. 5. Comparison between IOA, sNIOA and sNIOA-GM.

E. Application Case 2

Assume that there are w robot stations. For each station, g

heterogeneous robots are expected to be available. The total

number of robots is q wg . All of the robots are labeled with

number 1 to q . There are s tasks scattered on an area waiting

for execution labeled with number 1 to s . Each task needs to be

executed by one robot. One robot can perform multiple tasks in

turn. The goal is to assign robots to perform tasks and design

the execution sequence of each robot for the sake of minimizing

total energy consumption, which is mainly related to the

travelling distances of robots. The concerned context can be

proved to be an NP-hard problem [45]. The mathematical

model is built as follows.

Let A denote the th robot station. Given a set T , T is

the number of members in T . A robot ir is one of the robots

departing from station A to accomplish a sequence of

assigned tasks 1 2, , , iT

i i i iT t t t to ir and return to the same

place after finishing its jobs. Its destination set

, ,i iD A T A is labeled with number 1 to iD . uv is

taken as the distance between destinations u and v . uv is

defined as the binary variable representing the relationship that

the path goes from u to v . Then the concerned MRTA

problems can be formulated as follows:

1 1 , 1

mini iD Dq

uv uv

i u v u v

f

(9)

s.t. 1

q

ii

T T

, (10)

0,1 , , vuv u , (11)

1i iA v uA . (12)

For solving the above-mentioned problem by an

immune-inspired method, an antibody is initialized as a vector

whose length is s as shown in Fig. 6. Each position of the

antibody represents the label of a robot being assigned to

accomplish the task. One robot can be allocated to multiple

tasks.

Task ID

Antibody 5 1 q� � 3

Robot ID

1 2 3 4 � s

Fig. 6. An antibody of sNIOA-GM for MRTA.

For this MRTA problem, the procedure to compute the total

cost is given as Algorithm III. The population size is N . In

order to obtain the sequence that a robot performs a set of tasks,

the Simulated Annealing Algorithm [46] is used to design a

shortest path and its number of iterations is L . Therefore, the

evaluation process needs O N q L , which is

approximately equal to 2O N . Comparing with O N ,

which a simple objective function takes, it is reasonable to

expect that the computational cost can be decreased when the

time of fitness evaluation is shortened in this problem.


10

ALGORITHM III. FITNESS COMPUTATION FOR APPLICATION CASE 2

1: Population size is N ;

2: For 1:i N

3: 0Total distance ;

4: Collect assigned tasks for each robot;

5: For 1:j q

6: Compute the shortest path with use of Simulated Annealing algorithm

(Number of iterations is L );

7: Total distance Total distance Shortest Path ;

8: End for

9: Return Total distance as the fitness value for thi antibody;

10: End for

There are three robot stations 1A , 2A and 3A in an

environment located on [0, 70], [50, 0] and [100, 30],

respectively. Three robots at each station are used to deal with

20 tasks scattered on a 100 100 area. 1

AC , 2

AC and 3

AC

denote the number of required robots from each station,

respectively. Multiple solutions and related data are given in

TABLE 6 and Section G of the Supplementary File. There are

different kinds of allocation ways with different numbers of

robots dispatched from different robot stations. According to

these results, decision makers can choose one from them

depending on their situations.

For example, in this application case, the number of robots is

three at each station. However, when a set of tasks arrives,

some robots may be charged or have been already dispatched to

execute other jobs. It results in the reduction of available robots.

In traditional approaches, managers collect data, build models

and compute solutions at that time. This process consumes

much time before starting execution and thus delays the

completion of tasks. Besides, it needs to be done every time as

long as tasks arrive. With the use of our method, a group of

strategies covering different cooperative ways between robot

stations and execution paths can be obtained previously

according to the original information that each station has three

robots. In such circumstances, decision makers can easily

choose one from them regardless of the number of real-time

available robots.

Furthermore, it is also helpful to avoid complex modeling of

robot energy. For example, considering TABLE 6 and Section

G of the Supplementary File, if a robot in station 2A has

insufficient power to finish a large-scale set of tasks, strategies

(c), (e) and (f) are more appropriate than (a) even though the

latter one has the shortest path length.

TABLE 6

EXPERIMENTAL RESULTS OF APPLICATION CASE 2

No. Path Length 1

AC 2

AC 3

AC

(a) 476.36 1 1 1

(b) 520.44 1 2 1

(c) 535.55 1 1 2

(d) 555.00 1 2 2

(e) 555.34 2 1 1

(f) 567.40 2 1 2

(g) 570.55 0 2 2

(h) 573.72 1 0 3

(i) 604.00 2 2 2

(j) 610.67 2 2 1

V. CONCLUSION

In this work, an optimization strategy to provide multiple

solutions with same or similar quality is proposed for MRTA

problems. First, in order to decrease time of fitness evaluation

and accelerate computation, sNIOA is designed. The new

population is prejudged by the regression model such that

non-ideal antibodies can be eliminated before entering their

evaluation process and unnecessary computations are thus

avoided. Then, the GM operator inspired by the base pair is

proposed. Genes of an antibody are divided into several gene

groups. When a gene mutates, others in the same gene group

may mutate with a higher probability. The GM operator is

helpful to find preferable solutions when the number of

iteration is limited and the scale of a MRTA problem is

large-sized. To verify the effectiveness and efficiency of the

proposed algorithms, this work conducts a series of

experiments on test functions and compares them with several

heuristic algorithms. The results demonstrate the high speed of

sNIOA and search ability of GM. Furthermore, application

cases show that the proposed sNIOA-GM can well handle

MRTA problems by providing multiple solutions.

Different from other task allocation methods, the major

advantages of sNIOA-GM are: 1) it can provide multiple

solutions with same or similar quality for decision makers to

select and switch plans whenever such needs arise; 2) because

of the pre-judgment by a regression model, sNIOA is able to

reduce the evaluation time and computational cost; 3) the GM

operator reveals a potential connection between genes and

strengthens the optimization ability of sNIOA.

The further studies may include a selection mechanism and

switching strategy given multiple candidate solutions. Besides,

considering the balance between multimodal optimization

ability and uniformity of solutions, the proposed sNIOA-GM

still has some room for further improvement. Furthermore,

preserved individuals should be judged more accurately to

obtain different mutation probabilities according to their fitness

values.

ACKNOWLEDGEMENT

We dedicate this paper to Prof. Yongsheng Ding, who

proposed the original idea, provided technical assistance and

had helpful discussions with us. We feel very sad for Prof.

Yongsheng Ding’s unexpected passing away and miss him so

much.

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12

IEEE Transactions on Cybernetics, vol. 47, no. 11, pp. 3658-3668, Nov.

2017.

Li Huang (S’17) received the B.S. degree from

Donghua University, Shanghai, China, in 2013, where

she is currently pursuing the Ph. D. degree in control science and engineering. She is now a visiting student

with the Helen and John C. Hartmann Department of

Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ, USA, supported by

the China Scholarship Council. Her research interests

include cooperative control and dynamic optimization of multi-robot systems and intelligent automation.

Dr. Yongsheng Ding (M’00-SM’05) is currently a Professor at College of Information Sciences and

Technology, Donghua University, Shanghai, China. He

obtained the B.S. and Ph.D. degrees in Electrical Engineering from Donghua University, Shanghai, China

in 1989 and 1998, respectively. He is a Changjiang

Distinguished Professor appointed by the Ministry of Education, China. From 1996 to 1998, he was a Visiting

Scientist at Biomedical Engineering Center, The

University of Texas Medical Branch, TX, USA. From February 2005 to April 2005, he was a Visiting Professor

at Department of Electrical and Computer Engineering, Wayne State

University, MI, USA. From September 2007 to February 2008, he was a Visiting Professor at Harvard Medical School, Harvard University, MA, USA.

He serves as Senior Member of Institute of Electrical and Electronics Engineers

(IEEE). He has published more than 300 technical papers, and ten research monograph/advanced textbooks. His scientific interests include computational

intelligence, network intelligence, intelligent Internet of things, big data

intelligence, and intelligent robots.

MengChu Zhou (S’88-M’90-SM’93-F’03) received his B.S. degree in Control Engineering from Nanjing

University of Science and Technology, Nanjing, China

in 1983, M.S. degree in Automatic Control from Beijing Institute of Technology, Beijing, China in 1986, and Ph.

D. degree in Computer and Systems Engineering from

Rensselaer Polytechnic Institute, Troy, NY in 1990. He

joined New Jersey Institute of Technology (NJIT),

Newark, NJ in 1990, and is now a Distinguished Professor of Electrical and Computer Engineering. His

research interests are in Petri nets, intelligent automation, Internet of Things, big data, web services, and intelligent

transportation. He has over 700 publications including 12 books, 400+ journal

papers (300+ in IEEE transactions), 11 patents and 28 book-chapters. He is the

founding Editor of IEEE Press Book Series on Systems Science and

Engineering and Editor-in-Chief of IEEE/CAA Journal of Automatica Sinica.

He is a recipient of Humboldt Research Award for US Senior Scientists from Alexander von Humboldt Foundation, Franklin V. Taylor Memorial Award and

the Norbert Wiener Award from IEEE Systems, Man and Cybernetics Society

for which he serves as VP for Conferences and Meetings. He is a life member of Chinese Association for Science and Technology-USA and served as its

President in 1999. He is a Fellow of International Federation of Automatic Control (IFAC), American Association for the Advancement of Science

(AAAS) and Chinese Association of Automation (CAA).

Dr. Kuangrong Hao is currently a Professor at the

College of Information Sciences and Technology, Donghua University, Shanghai, China. She obtained her

B.S. degree in Mechanical Engineering from Hebei

University of Technology, Tianjin, China in 1984, her M.S. degree from Ecole Normale Supérieur de Cachan,

Paris, France in 1991, and her Ph.D. degree in Mathematics and Computer Science from Ecole

Nationale des Ponts et Chaussées, Paris, France in 1995.

She has published more than 100 technical papers, and three research monographs. Her scientific interests include machine vision,

image processing, robot control, intelligent control, and digitized textile technology.

Yaochu Jin (M’98-SM’02-F’16) received the B.Sc., M.Sc., and Ph.D. degrees from Zhejiang University,

Hangzhou, China, in 1988, 1991, and 1996 respectively, and the Dr.-Ing. degree from Ruhr

University Bochum, Germany, in 2001. He is a

Professor in Computational Intelligence, Department of Computer Science, University of

Surrey, Guildford, U.K. He was also a Finland Distinguished Professor funded by the Finnish

Agency for Innovation and a Changjiang

Distinguished Visiting Professor appointed by the Ministry of Education, China. His main research interests include data-driven surrogate-assisted evolutionary

optimization, evolutionary multi-objective optimization, evolutionary learning,

interpretable and secure machine learning, and evolutionary developmental

systems. Dr. Jin is the Editor-in-Chief of the IEEE TRANSACTIONS ON

COGNITIVE AND DEVELOPMENTAL SYSTEMS and Co-Editor-in-Chief of Complex & Intelligent Systems. He is the recipient of the 2018 IEEE

Transactions on Evolutionary Computation Outstanding Paper Award, the 2015 and 2017 IEEE Computational Intelligence Magazine Outstanding Paper

Award. He is a Fellow of IEEE.

Documents

Multiple solution Optimization Strategy for Multi …epubs.surrey.ac.uk/848610/1/SMCA-18-04-0423-Clean.pdfMultiple-solution Optimization Strategy for Multi-robot Task Allocation Li